Indecomposable projective modules on affine domains

C ^OMPOSITIO M ^ATHEMATICA

V. S ^RINIVAS

Indecomposable projective modules on affine domains

Compositio Mathematica, tome 60, n

^o

1 (1986), p. 115-132

<http://www.numdam.org/item?id=CM_1986__60_1_115_0>

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115

INDECOMPOSABLE PROJECTIVE MODULES ON AFFINE DOMAINS

V. Srinivas

e Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands

Let A be ^an affine domain i.e. an

integral

domain of finite

type

^{over an}

algebraically

closed field k. If A has Krull

dimension d,

^then

by

^the

stability

theorem of Bass

[Ba],

any

indecomposable projective

^A-module

has rank at most d. In this paper, ^we consider the conditions under which A has

indecomposable projective

A-modules of rank d. Our main result is:

THEOREM: Let k be a universal

domain,

^{and assume that d = dim} A 3.

Assume that

Spec

^{A has at worst} isolated normal

singularities; further if

dim

A = 3,

^then

char,

^{k =1=}

2, 3,

5. Then there exist

indecomposable projective

^A-modules

of

^{rank d}

if

^and

only if F0K0(A) ~

^0.

Here

F0K0(A)

^{is the}

subgroup

^{of the} Grothendieck _{group of}

projec-

tive A-modules

generated by

^the classes of residue fields of smooth

points;

^{it is}

isomorphic (upto 2-torsion)

^{to the} Chow group of ^zero

cycles

of

Spec

^A

(see §1

^{for the}

definition).

The restriction to

char,

^{k =1=}

2, 3,

^{5 if}

d ⁼ 3 is because we need to use the theorem of resolution of

singularities

for

3-folds,

^{which is}

proved

under the above restriction in

[A].

^{In a}

footnote in their paper

[MK],

^M.P.

Murthy

^and Mohan Kumar state that

Abhyankar

^{can now} prove resolution for 3-folds in all

characteristics;

assuming

^{this we} may

drop

^the

hypothesis

^{on the} characteristic.

The

hypothesis

^{on the}

singularities

is needed for technical reasons in the

proof,

but may be unnecessary: 1 do not know any ^counter

example

for non-normal A or for normal A of dimension 3 with non-isolated

singularities. ^However,

^{it seems}

plausible

^{that k must be a}

’big’

^{field in}

order that

F0K0(A)

^be

non-zero; I

^{do not know an}

^example

^{of an affine}

domain of

dimension

^{2 over} Qu with

F0K0(A) ~

^0.

Many examples

exist of such affine domains over the

algebraic

^closure

of Q(x, y ) (x, y indeterminates)

^with

F0K0(A) ~

^{0. But our} methods do not seem to be

directly applicable

^{to such}

rings

^{*. We discuss}

examples

^of

rings

^with

F0K0(A) ~

^{0 in}

^§4

^and

^§5.

The

proof

^{is based on the}

following

^idea.

Suppose

^{P is}

projective

^of

rank d ⁼ dim

A 2,

^{and for a} suitable notion of the Chow

ring together

* One can combine our result with the method of [BS] ^to get ^such results.

with a

theory

^{of Chern}

classes,

^{we have}

C1(P) =

^...

Cd-1(P)

^{= 0. Then}

if

P = Q

^{~ L where L E Pic}

A, rank Q

^{= d -}

1,

^{we have an}

identity

ⁱⁿ

the Chow

ring

This forces

Ci(Q) = (-1)iC1(L)i,

^and

Cd(P) = (-1)d-1C1(L)d.

^Thus

our Theorem will follow once we show that if

F0K0(A) ~ 0,

^{then there} exists a

projective

^{module P with}

Ci(P) = 0

^for

^{i d,}

^{and such that}

Cd(P) ~

^{0 is not}

rationally equivalent

^{to a}

cycle

^{of the form}

(-1)d-1C1(L)d.

The idea of the

proof

^{is to} show that if

F0K0(A) ~ 0,

then the Chow group of

0-cycles CHd(A)

is ’infinite dimensional’ in a

suitable sense, while the classes of

cycles

of the form

(-1)d-1C1(L)d

^lie

in a countable union of

’algebraic

^{sets’ in}

CHd(A),

^each

component

^of which has dimension bounded

by

^{a number}

depending only

^A

(this

^relies

on the

theory

of the Picard

variety).

This is done

using

ideas of Roitman

[R]

which have to modified somewhat to deal with

affine, possibly singular

^varieties

(Roitman basically

deals with smooth

projective

^varie-

ties).

Roitman’s ideas

directly apply

^{when A} is smooth of dimension

2,

but ^our modifications seem to be needed even for smooth A if dim A = 3.

Finally,

^{we do not} know if the theorem

ought

^{to be true if dim}

A = 4, though

^{we know no counter}

examples.

^{If we}

try

^an

analogous argument,

and let P be

projective

^{of rank 4 with}

Cl ( P )

⁼

C2 ( P )

⁼

C3 ( P )

⁼

0,

^then

suppose that ^P

= P1 ~ P2

^with

rank Pi

^{= 2. A} Chern class

computation yields C1(P2) = - C1(P1), C2(P2)

⁼

C1(P1)2 - C2(P1),

^and

C4(P)

= 1 2C1(P1)4 - C2(P1)2.

^The

0-cycles

of the form

C1(P1)4

range ^{over a} countable union of

algebraic

varieties of bounded dimension.

However,

CH2(A)

could be infinite

dimensional,

so we seem to have no control on

the set of classes of the form

C2(P1)2 in ^CH4(A).

This

problem

^was

suggested

^to the author

by

^M.P.

Murthy,

^{in the case}

d = 2. I wish to thank N. Mohan Kumar and M.V. Nori for

stimulating discussions;

ⁱⁿ

particular,

Mohan Kumar

suggested

the trick of

looking

at

projectives

^with

Ci

^{= 0 for i}

^d,

which reduces the

problem

^{to line}

bundles ^even if d = 3.

§1. Equivalence

^{relations on zéro}

cycles (af ter Roitman)

Let X be a normal

projective variety

^{over an}

algebraically

closed field

k,

and let U c X be the set of smooth

points.

The group of ^zero

cycles

^on

X,

^denoted

by Z0(X),

^{is defined to} be the free abelian _group ^on the

points

^{of U. Let}

R(X)

^c

Z0(X)

^{be the}

subgroup

^of

Zo ( X) generated by cycles

of the form

(f ) c

where C c X is a curve with

C ~ (X - U) = Ø

and

f

^{is a non-zero} rational function on

C; (f)C

denotes the divisor of

f

on C

(note

^that

C ~ ( X - U ) = Ø just

^{means C c}

^U;

^{we write it as above}

to

emphasise

that C is a closed subset of

X).

The Chow group of

0-cycles

is defined to be the

quotient Z0(X)/R(X);

^{we denote it below}

by A0(X) ^(rather

^than

CHd(X),

where d = dim

X). Similarly

^{if A is a}

normal affine

domain,

^{let V c}

Spec

^{A be the set of smooth}

points, Zo ( A )

the free abelian _group ^on

points

^of

V,

^and

A0(A)

^the

quotient

^of

Z0(A)

modulo the group

generated by (f)C

^where

C c Spec A

^{is a curve,}

C ~

((Spec ^{A) -} V) = Ø,

^and

f

^is a non-zero rational function ^on C.

If

X,

^{U are as}

above,

^{the n th}

symmetric product sn(u) parametrizes

effective zero

cycles

^of

degree n

^{on X i.e.}

cycles 03A3ni(Pl)

^with

ni &#x3E; 0,

03A3ni

= ^n,

P, e

^{U. There are} natural maps

where

[ A denotes

^{the zero}

cycle corresponding

^{to A E}

Sm(U).

^In

[S1],

Lemma

(1.1)

^we showed that if

is induced

by ’¥ m n’

^then

03B3-1m,n(0)

^{is a countable union of}

^locally

^closed

sets in

Sm(U) Sn(U),

and used this to

study

^’infinité

dimensionality’

of

A0(X).

^{Here we}

^study

^infinite

dimensionality

^for

cycle

groups associ- ated to more

general equivalence

^relations.

Let

W ~ Z0(X)

^{be a}

subgroup.

Then W defines an

equivalence

relation on

0-cycles, namely [A] ~ [ B iff [A] - [ B ] ~

^{W. Let}

AW0(X) = Z0(X)/W

^{be the}

corresponding quotient.

^{We define W to} be admissible if

(i)

^{W ~}

R(X) (ii)

for any ^{m, n} if

crm,n: Sm(U)

^X

Sn(U) ~ AW0(X)

^is

induced

by ’¥ m,n’

^then

~-1m,n(0)

^{is a} countable union of

locally

^closed

sets. *

We recall some

terminology

of Roitman. For ^a

variety X

^{over a}

universal

domain,

^{a c-closed set in X is a} countable union of closed subsets of X. For any c-closed ^{set F =}

U Fi

ⁱⁿ

^X,

^we define dim F = sup

(dim Fi).

^This

concept

^{is well}

defined,

^and

depends only

^{on the set F}

and not on the

specific decomposition

^{into sets}

Fi ^(since

^an irreducible

variety

^{over a} universal domain is not the union of a countable set of proper

subvarieties). Similarly,

^a

c-locally

^{closed set in X is a countable}

union of

locally

closed subsets of X. For any

c-locally

^closed

set

^{G =}

UGi

with

Gi locally ^closed,

^{we define} the c-closure of G to be

UGil

^{This is}

also characterised as the smallest c-closed subset of X

containing ^G,

since if F =

U Fi :)

^{G where}

F

^are

^closed,

^{then each}

^Gi

^{is contained in a}

finite

union of the

F,

^{which is a closed set}

^(and

^{so contains}

^Gi).

^In

particular

the c-closure of G does not

depend

^{on the}

decomposition

^{of G}

* By the methods ^of [LW] ^{one can} prove that

~-1m,n(0)

^{is a countable union of closed sets if}

W is admissible, but we do not need this.

into

locally

^{closed subsets. For} any

c-locally

^{closed set}

G,

^{we define} dim G to be the dimension of its c-closure.

Finally,

^a c-open ^set in X is defined to be the

complement

^{of a c-closed set.}

We have the

following lemma, essentially

^{due to Roitman.}

LEMMA

(1.1):

^{Let X be a}

variety

^{over a universal}

domain,

^{r c X X X the}

graph of

^an

equivalence

^{relation on X.}

Suppose

^{that r is}

c-locally

^{closed in}

X X X. Then there is a c-open ^set V c X and an

integer ^d

0 such that

for

any ^{x E}

V,

^{the set}

is a

c-locally

closed subset

of

^X

of

^dimension

d,

such that all irreducible components

of

the c-closure

of rx

^have dimension d.

(We

^{note that} every c-closed set in X has ^a

unique

irredundant decom-

position

into irreducible closed sets, ^{so it makes sense to} talk of the

components

^{of a c-closed}

set).

PROOF: Let

r

be the c-closure of r. Then

r

is also the

graph

^{of an}

equivalence

^{relation on X. One}

_sees easily

^{that there}

exists a

^{c-closed set}

F ~ X such that

for x ~ X - F, (0393)x = {y ~ X|(x, y) ~ 0393} is just

^the

c-closure of

0393x.

^{Hence we reduce to the case} when r itself is

c-closed,

and so

rx

is also c-closed. Let r ⁼

U0393i

^{be an} irredundant

decomposition

into irreducible closed subsets of X X X.

We claim first that there exists a

unique component

^{of r}

containing

the

diagonal

^{in X X X.}

Indeed,

^{there is a} proper c-closed subset ^{F c X} such that for all x

E X - F,

^either

(i) ({x} 0393x) ~ 0393l is empty, or

(ii)

^if

0393i,x

⁼

^{{y ~} X|( x, y ) E ]ri 1,

^then

fi x

^{is a finite}

^disjoint

^{union of}

irreducible subvarieties of dimension

equal

^to

(dim rl -

^dim

^X);

further

rx = U0393i,x is an

irredundant

union,

^{where i runs over}

indices such that

]ri @ x :gÉ 0 (by irredundancy

^{we mean that no}

component

^of

rl,x

is contained in

0393J,x ^{for j ~ i).}

Now

suppose y E 0393i,x - U 0393J,x -

^{F. Then}

( x, y ) E ]ri - U rj.

^For

any y E rx,

^{we have}

ry

⁼

^lx

^{as r is an}

equivalence

^{relation. If}

r y = U0393i’,y

is the above

decomposition (which

^exists

because y 5É F),

^{then the}

component

^of

0393l,x containing y

^{lies in a}

unique 0393i’,y,

^{so that}

( y, y) E fi,

-

U0393J.

^Thus

^rt,

^{is the}

^{unique component}

^{of r}

containing

^the

diago-

j 4= i,

nal.

Further, Fi,x

^and

r,,

^are

equidimensional

schemes with ^a common

irreducible

component (namely,

^the

component through y )

^{and so}

dim

rl = dim 0393i,x + dim

^{X = dim}

0393i’,y

^{+ dim X = dim}

0393’i.

Since i was

arbitrary,

^dim

rl

^is

independent of i,

^at least for all

components fi

^which dominate X under the first

projection.

^Thus

dim

fi,x

^{= dim}

0393J,x

for all such

fi, 0393j

^i.e.

^rx

^{is a union of} irreducible varieties each of which has dimension d ⁼

(dim Fi,

^{- dim}

^X).

This proves the lemma.

Now let W be an admissible

equivalence

^{relation on}

0-cycles ^{on X,}

where X is ^a normal

projective variety

^{over a} universal domain. Then via ~m,n, ^{W induces a}

c-locally

^closed

equivalence

^{relation on}

Sm(U)

^X

sn(u),

^{where U = X -}

Xsing.

^Let

^{W m, n}

be the number d

given by

^Lemma

(1.1)

^{in this} context, ^{and let}

We define

SD(U)

^{to be a}

point, corresponding

^{to the zero}

cycle 0,

^{and let}

dn

⁼

dn,o,

^{wn =}

Wn,o

^etc.

PROPOSITION

(1.2) :

^{There are}

integers dW, jW

^and no &#x3E; 0

depending only

on X and W such that

0 dW(X)

^dim

^{X, and for}

^m

+ n n0

^{we have}

PROOF: We first show that

Wm,n = wm+n. Indeed,

^consider

Clearly ~m,n((A, ^{B)) =} crm,n((C, ^D))

^iff

^~m+n(A

⁺

^D)

⁼

~m+n(B

⁺

C)(~m+n

⁼

~m+n,0).

Now rm,n is the

graph

^{of the}

equivalence

relation determined

by _~m,n,

^{so that if}

0393m,n0

^{is the}

unique component

^of the c-closure of

0393m,n containing

^the

diagonal,

^then

Let 0393m+n ~

Sm+n(U)

^X

Sm+n(U)

^{be the}

graph

^{of the}

equivalence

^rela-

tion determined

by

~m+n, and let

0393m+n0

^{be the}

unique component

^{of the} c-closure

containing

^the

diagonal;

^then

Let NF:

Sm(U)

^X

Sn(U)

^X

Sm(U) Sn(U) ~ Sm+n(U)

^X

sm+n(u)

^be

given by (A, B, C, D) - ( A + D,

^{B +}

C).

Then 03A8 is

finite,

^and

03A8-1(0393m+n)

⁼ 0393m,n. Hence dim

0393m,n0 =

^dim

0393m+n0 (since 03A8

maps the

diagonal

^{to the}

diagonal)

^{i. e.}

_Wn,n

⁼ wm+n, ^and

dm,n = dm+n.

Let Vn ~ U

Sn(U)

^{be the set}

Similarly

^{for A E}

sn(u)

^let

VAn

^be

(the image

^{in U}

of)

^{the fiber over A of}

the

projection

^{Vn ~}

Sn(U).

For any ^n, V" is a

c-locally

^{closed set in}

U X

Sn(U),

^{and for} any

A, VAn

^{is a}

c-locally

^{closed set in U. Thus for}

some

non-empty

c-open ^set

U1

^~

Sn(U)

^{and for all A E}

Ul,

^{we have}

Let 03B4n

^{= dim}

hA

^for

any A E UI.

For any A E

Sn(U)

and any ^{x E} U ^we have

VnA ~ Vn+1A+x,

^{so that}

03B4n 03B4n+1 dim X.

^Thus

03B4n = 03B4n+1

^{for all}

n n0. Let 03B4 be this ^common

limiting value,

^{and let}

dW =

^{dim X - 03B4.}

Consider the map

(addition)

If

0393n+1A = {B ~ Sn+1(U)|(A)=~n(B)}

^for

any A ~ Sn+1(U),

^then

p1(03C0-1(0393n+1A))

⁼

Vn+1A,

where p1 is the

projection

^{U X}

Sn(U) ~

^{U. For}

x E

Vn+1A,

the fiber of

(p1)| 03C0-1(0393n+1A) ^{just 0393nB,}

where B is the effective

0-cycle A-[x].

Since w n ⁼ dim

0393nB

^{for B in some c-open subset of}

^U,

^and

similarly wn+1 = dim 0393n+1A

^for

general A,

^{we have}

wn+1 = wn + 03B4

^i.e.

dn+1 = dn

⁺

dW ^{for n}

n o . This proves the

proposition.

REMARK: This

proof only

^{used the}

property (ii)

^{in the} definition of an

admissible

equivalence

^relation.

LEMMA

(1.3):

^Either

d w = 0 or dW 2. ^If dW = 0, then for

^{some n}

image(~n,n) is a ^subgroup of AW0(X)

^with

cyclic quotient.

PROOF:

Suppose dW

^{1. Then}

^{for n}

n0, ^we have dim

Vn+1A

^{dim X}

- 1 for

any A E Sn+1(U), by

definition of

dX =

^{dim X - 8. Thus there}

exists a smooth curve C c X with

C n (X - U) = Ø

^{and Cn}

Vn+1A ~ Ø

for all

A ~ Sn+1(U) (in

fact any

general

^smooth

complete

intersection

curve C will

do).

^Then for any A E

Sm(U)

^{with m} &#x3E; n0 ^{we can} find

Al

^E

Sn0(U)

^and

A2 E sm-no(C)

^{such that}

Since W is ^an admissible

equivalence relation,

^{we have}

diagrams

^{for all n}

If we choose a base

point Po

^E

C,

^and g ⁼ genus

(C),

^then

by

^the

Riemann-Roch

theorem,

for n &#x3E; g and

any B ~ Sn(C),

^{there exists}

B 1 E S g( C )

^{such that}

Thus for any m &#x3E; n o ⁺ g, and

any A E ^Sm(U),

there exists B ~

Sn0+g(U)

such that

Thus

image _{~m,m =} image ~n0+g,n0+g

^{f or}

m &#x3E; n0 + g

^Le.

dX = 0

^and

image

~m,m is a

subgroup

^of

AW0(X)

^whose

^quotient

^{is the}

^cyclic

^group

generated by

^the

image of ~1(P0).

We recall another définition. _{For any}

variety Z,

^{a set} theoretic map

Z ~ AW0(X)

is defined to be ^a

regular

map if there exists a

diagram

where

f,

g are

morphisms

^and

f

^is

surjective.

^{Since W is}

admissible,

^if

C c X is ^a smooth ^curve with

C ~ (X - U) = Ø,

the natural homomor-

phism

^{Pic C -}

AW0(X)

^{is a}

regular

map. Given any finite set of

points

ⁱⁿ

U,

^{there is a smooth curve C as above}

through

^these

points.

Thus any

given

^zero

cycle

^{is in the}

image

^{of such a}

regular homomorphism.

^Since

the jacobian J(C)

^is

divisible,

^if

AW0(X)’ ~ AW0(X)

^{is the}

subgroup generated by cycles

^of

degree 0,

^then

AW0(X)’

^is divisible.

Clearly ~n,n

factors

through AW0(X)’.

PROPOSITION

(1.4):

^Let

dW = 0.

Then there is a

surjective regular

^homo-

morphism f : A ~ AW0(X)’,

^{where A is an abelian}

variety

^{which is a}

quotient of Alb(X), and f

has countable kernel.

PROOF: We first remark that there is ^a

surjective regular homomorphism f-: 4 ~ AW0(X)

^{from an abelian}

variety A,

^{such that ker}

f

is countable.

This is

proved

^for

W = R(X) (i.e.

^rational

equivalence)

ⁱⁿ

[SI],

^Lemma

(1.3).

^{But the}

only property

^of

R(X)

used in the

proof

^{is that it is} admissible in our sense.

Given this

homomorphism,

^{choose a base}

point P E U,

^{and let} r c U X A be the set

Then r is

c-locally closed,

^{and since}

f

^is

surjective,

^r

surjects

^{onto U}

under the

projection.

^{Hence some component}

ro

^{of the c-closure of r}

dominates U. Let

rl

⁼

ro

ⁿ

^r;

^then

03931 ~

^{U has}

countable,

^{and hence}

finite,

^fibers

(since f

^{has countable}

fibers).

^{There is an} open ^set

Uo

^{c U}

such that if

r2

⁼

rl X U Uo,

^{then 03C0 :}

r2

^~

Uo

^{is a} finite flat

morphism

^of

varieties. Let n be the

separable degree

^and

p m

^the

inseparable degree

^of

this

morphism.

^{If x E}

Lo

^is

general,

^then

03C0-1(x) = {(x, a1),..., ^(x, an)}

f or n distinct

points

a1,...,an ^{E A .} Then

x ~ pm(a1

^{+ ···}

+ an) gives

^a

well defined

morphism 03A8 : U0 ~ A

^{such that}

f 0 ’l’: U0 ~ AW0(X)’

^{is the}

natural map

crl,l multiplied by n. pm.

^Hence

f o 4,(U.) ^generates AW0(X)’

(since AW0(X)’

^is

divisible,

^and

~1.1(U0 {P}) ^generates AW0(X)’).

^Since

f

has countable

fibers, 03A8(U0) ^generates

^A.

Regarding ’1’

^{as a rational}

map

X ~ A, 03A8

^factors

through Alb(X) by

the universal

property

^{of the} Albanese

variety.

Since the

image

^of

Alb(X) ~ A

^contains

41(U), Alb(X) ~ A

^is

surjective.

We

apply

these results to the

following

situation. Let X be a normal

projective variety

^{over a universal}

domain,

^{and let V ~ X be an} open subset

containing Xsing.

^{Then we have an exact sequence}

where

A0(V)

is the free abelian group ^on

points

^of

V - Vsing

^{modulo the}

subgroup generated by cycles (f ) c

where C ~ V is a

(closed)

curve, C ~

Vsing ^{= Ø,}

^and

^f

^is a non-zero rational function on C. The above

presentation

^for

Ao(h)

^follows

because

⁼

Xsing.

^{From the}

^presenta-

tion,

^{if we set}

AW0(X) = A0(V),

^{then W is an} admissible

equivalence

relation.

LEMMA

(1.5): Suppose

^{dim X}

2, dW

^{= 0 and X - V generates Alb X.}

Then AW0(X) = 0.

PROOF; Clearly AW0(X) = AW0(X)’,

^{so it suffices to} show that if

f : A -

AW0(X)’

^{is a}

surjective regular homomorphism

with countable

kernel,

then A ⁼ 0. As in the

proof

^of

Prop. (1.4),

^we claim there is a

diagram

A

priori

^we

only

^{have such a}

diagram

^with

Uo

instead of

U,

^{but the} rational

map X -

Alb X with P ~ 0 is ^a

morphism

restricted to U. Thus

we would have two

regular

maps

U ~ AW0(X)’, namely n.p.m ~1,1

^and

f -

^{h °} g, which agree ^on

Uo.

^We claim that

they

^must agree

everywhere

on U.

Indeed,

^{if C c U is a}

complete

^{smooth curve with}

C ~ U0 ~ Ø,

^then

we have 2

regular

maps ^{C -}

AW0(X)’ ^namely npm~1,1 | C

^and

^f

^{° h °}

g|C.

Both induce

regular homomorphisms

^Pic

C ~ AW0(X)’,

the former be-

cause W is

admissible,

and the latter because

Pic°C

⁼ Alb

C,

^{so that}

g c:

C ~ A induces a

homomorphism

^{Pic C ~ A. These two maps Pic C}

~

AW0(X)’

^are

^equal,

^{because Pic C is}

generated by

the classes of

points

in C ~

Uo.

^{Thus the two}

original

maps ^on U _agree on C. But such a curve

C can be found

through

any

point

^of

U,

since X is normal. ^*

Now X -

Vc U,

^and

f

^{o h o}

g(X - V)

^{= 0 in}

AW0(X) (by

^definition

of

W,

^{and the}

equality

^{of the 2}

maps).

^{Thus h o}

g(X - V) ~ (ker f)

which is a countable

subgroup

^{of A i.e. h o}

g( X - V)

is finite. But

g(X - V)

^{c Alb X}

generates Alb X, by

^choice

of V,

and Alb X ~ A.

Hence A = 0 i.e.

AW0(X)-

^{= 0.} This proves the lemma.

§2.

Families of

0-cydes

associated to Picard families

Let X be a normal

projective variety

of dimension d over an

algebrai- cally

closed field

k,

^{and let T be a normal}

variety.

^Assume

given

L~

Pic(X T),

and for each t ~

T,

consider the rational

equivalence

class of the zero

cycle (-1)d-1Ci(Lt)d ~ A0(X),

^where

Lt = L|X {t}.

Here if X is

smooth,

the intersection

theory

is the usual ^one

(see

^Fulton

[F])

^{while if X is}

singular,

^{we use the}

theory

^{of Levine}

[L1] (we

^discuss

this

theory briefly

ⁱⁿ

§3).

^{In this}

section,

^the

only property

^{of the} intersection

theory

needed is that if

D1,..., Dd

^are effective Cartier divisors on X

meeting properly,

^and

r1 supp Dl

^c

^U,

^then

D1

^{~ ...}

~Dd (as schemes) represents

the rational

equivalence

^{class of}

[D1][D2] ... [Dd]

in

A0(X). Returning

^{to our}

situation,

^{we have a} map of ^sets cp: T ~

Ao(X), given by t - (-1)d-1C1(Lt)d.

LEMMA

(2.1):

Under the above

conditions, ~ :

^{T ~}

A0(X) is a ^regular

^map

in

the following

^{sense: there exist}

a finite

open

cover {Ui} of T, positive integers

mi, n,

and morphisms 1: : Ui ~ Smi(U)

^X

Sni(U) giving

^commuta-

tive

diagrams

(where _03B3mi,ni

^are the natural

maps).

* We could have used the c-closedness of W-equivalence ^to directly prove the equality ^of the 2 maps; but the proof ^of c-closedness is more subtle.

PROOF: Since T is

quasi-compact,

^{we need}

only

prove the

following:

for each t E T there exists a

neighbourhood Ut of t, positive integers

^m,

n and a

morphism ft : Ut ~ sm(u)

^X

Sn(U)

^{such that}

cr 1 Ut

⁼

^{Ym, n}

^°

^ft.

Let .Yé’E Pic X be

ample.

^Then

p*1H~ Pic(X T)

^is

relatively

^am-

ple

^for

p2 : X T ~ T.

^{Hence for}

large n,

^we may ^assume that Hn and

Lt ~ Hn

^{are very}

ample, Hi(X, Hn) = 0

^for

i &#x3E; 0,

^and

(Rip2)*(L~p*1Hn)

^{= 0} for i &#x3E; 0.

By

^{the base}

change

^theorem

(see [Ml],

pg.

46)

^if

k(t)

^{is the} residue field of

(2t,T

^then

where

and j :

^{X X}

Spec Ot,T ~ X

^{T is the} natural map. Fix d effective di- visors

H1,..., Hd

in the linear

system | Hn |

^{on X such that}

Hl

^{n ...}

~ Hd

^{is a finite set of}

points

contained in

U;

ⁱⁿ

particular

^all

the

partial

intersections are proper i.e. dim

Hil

^{~ ...}

~ Hik = d - k

^for

any k

distinct indices

i1,..., ik. ^Next,

^{choose d divisors}

D1, ... , Dd

ⁱⁿ

1 | Lt~Hn ^|

^{on X} such that all the intersections

Di1 ~ ... ~ Dik n Hil

~ ...

Hjl

^are proper, and all ^zero dimensional intersection

cycles

^are

supported

^{on U}

(we

^can choose the

Di inductively, using

the remark that if

X ~ PN, Y1,..., Ym ~ X

subvarieties: then the

general hyperplane

section of X intersects all the

Y properly).

^Now

Ox(Di - Hi) ~ Lt,

^so

the intersection class

(-1)d-1C1(Lt)d ^{is just}

the class associated to the

zero

cycle

the

cycle

is well defined and

supported

^{on U}

by

choice. Let

sl, ... , sd E

H0(X, Lt~Hn)

be sections

corresponding

^to the divisors

D.

^From

the consequence of the base

change

theorem noted

above,

^{we can}

find ^a

neighbourhood

^{V of t ~ T and sections}

Si E H0(X V, L~p*1Hn | X|V)

^such

that si

^restricts

to sl on X {t}.

If

b,

is the divisor of

st,

^and

Iii

⁼

Hi ^{X V,}

^{then all the} intersections

DI1 ~ ... ~Dik ~ Hj1 ~ ... ~Hjl

^{are proper, at least on some} smaller open set of the form X

W, t ~

^{W c V}

(since

^{the dimension} of the fibers of a

surjective

proper

morphism

^of varieties is an upper semi-continuous function ^on the base

[Hl]

^{II Ex.}

3.22). By shrinking

^{W further we can}

ensure that for k + 1

= d,

all intersections

If

r = r(il, ... , ik; j1,..., jl) = (Di1 ~ ... ~ Dik ~ Hj1 ~

^...

~ Jl)

ⁿ

(X W),

^{where k + 1 =}

d,

^{then 0393 ~ W} is flat and proper of relative di-

mension 0

(flatness

^is

^immediate

from the Koszul resolution for the ideal sheaf of r ^{c X X}

W;

^note that r c U X W where U is

smooth).

^{Since W}

is

normal,

if 03A8 : r - W has

degree

^{q, then we have an induced mor-}

phism W ~ Sq(U) given by w ~ [03A8-1(w)]

^{as an effective}

0-cycle

^of

degree q

^{on U. Give two}

morphisms

g : W -

Sq(U),

^{h : W -}

sr(u),

^we

have a

morphism g + h : W ~ Sq+r(U)

^obtained

by

addition of

0-cycles, (g + h)(w) = g(w) + h(w).

Hence for suitable

positive integers

^{m, n we}

have ^a

morphism f W :

^{W -}

sm(u)

^X

Sn(U)

^{such that}

as

0-cycles,

^{where the zero}

cycle

A - B with

A,

^{B effective of}

degrees

^m,

n

corresponds

^to

(A, B) ~ Sm(U)

^X

Sn(U). Thus Ut = W, h = fW

^have

the

required properties.

COROLLARY

(2.2):

Let X be a normal

projective variety

^{over an}

algebrai- cally

^closed

field

k. Then there exists a countable

family of

^varieties

Wi, pairs of positive integers

mi, ni and

morphisms

gi :

wi ~ sm,(u)

^X

,Snt(U)

such

tha~ if fi = 03B3mi,ni ° gl, fl : Wi ~ A0(X),

^then

( i ) Ufi(Wi) ~ A0(X)

contains the rational

equivalence

^classes

of

^all

0-cycles ^of

^the

^form (-1)d-1C1(L)d,

^{cPE Pic}

^X,

^{d = dim X.}

( ii ) dim W = q is independent of

^i.

PROOF:

By

^{a result of}

Chevalley [C],

^{there is an abelian}

variety A

^and

E9OE

Pic(X A)

^{such that}

(A, Y)

^{is a Picard}

family

^{i.e. if}

^{Pic0X ~}

^{Pic X}

is the

subgroup

of bundles

algebraically equivalent

^to

0,

^{then the} map

A -

^{Pic X}

^{given by a ~ La}

^{= L}

|X {a} ^gives

^an

isomorphism

of groups

A - Pic°X; further,

any

family

^{of line bundles on X}

parametrized by

^a

normal

variety

^{T is induced}

by

^a

morphism

^{T -A. The} Neron-Severi group

NS( X)

^{= Pic}

X/Pic°X

^is

finitely generated (see

^Kleiman

[K],

^for

example).

^{Thus if}

f OE

^{Pic X}

^give

^a

(countable)

^{set of coset}

representa-

tives for Pic

X/PicoX,

^then

p*1Lj~L~ ^{Pic(X A) give}

^{a countable set}

of families of line bundles

on X,

^{each with}

parameter

space

A,

^such

that every line bundle ^{on X occurs}

exactly

^{once in a}

unique family.

Corresponding

^to each such

family,

^{we can associate a finite cover}

{Wij}

^{of A and}

^morphisms gij : Wij ~ SmiJ(U) XniJ(U)

^{such that}

VmiJniJ

^o

^gij(w)

^{is a zero}

cycle representing (-1)d-1C1(Lw)d,

^where

Lw = p*1Li~L|X {w}.

^{We can} reindex the

{Wij}i,j ^by

^the

^positive

integers, giving

^{a set}

{Wi}

of varieties with all the claimed

properties,

with

dim W

= q ⁼ dim

Pic°X.

§3.

Proof of the Theorem In fact ^{we can} prove ^a

slightly stronger

^result.

THEOREM: Let

X/k be a projective variety of

dimension

d

^{2 over a} universal domain with isolated normal

singularities.

^{Let V =}

Spec

^{A be an}

affine

open subset

of

^X

containing

^the

singular

^locus.

Suppose A0(V) ~

⁰

( equivalently FoKo(V)

⁼¹⁼

0).

Then there ^exist

(uncountably many) projec-

tive A-modules P

of

rank d such that P cannot be written as

Q

^{E9 L with}

LEPicA.

We note that this

implies

the theorem stated in the

introduction,

^since the

hypothesis

of that theorem

imply

^{that V =}

Spec

^{A can be}

compacti-

fied to a

projective variety X

^{such that}

Xsing

^{c V}

^(this

^uses resolution of

singularities).

^Thus

F0K0(V) ~

⁰

implies

the existence of

indecomposa-

ble

projectives. Conversely,

^if

FoKo(V)

^{= 0}

^{and d 3,}

^{then every}

projec-

tive module of rank d has a trivial direct summand i.e. a unimodular element. This is obvious for d ⁼ 1 and follows from the

Murthy-Swan

cancellation theorem if d = 2

[MS].

^If

d = 3,

this is the main result of

[MK],

^{if A is smooth over}

k;

it has been

generalized

^to

arbitrary

^normal

A in

[L2].

^{We note} that the conditions

F0K0(V) ~

^{0 and}

A0(V) ~

^{0 are}

equivalent -

^{indeed we have}

surjections 03A8 : A0(V) ~ F0K0(V)

^and

Cd : F0K0(V) ~ A0(V)

such that ’IF 0

Cd

^and

Cd 0 ’IF

^both

equal

^multi-

plication by (-1)d-1(d - 1)!;

but both groups ^are divisible.

PROOF OF THE THEOREM: In the notation of

§1,

^set

AW0(X) = A0(V).

Since X is

projective

and V ~ X is ^an affine _open

subset,

^{X - V}

generates

^{Alb X}

(if

^{Y c X is a}

general

2-dimensional linear space

section,

^then Alb Y - Alb X

[La],

^and

Y - (V ~ Y) is the complement

of ^an affine open ^set in Y which thus

supports

^an

ample

^{divisor on Y}

[H2]).

^Since

AW0(X) ~ ^0, by

^Lemma

(1.5)

^{we have}

dW

2 i.e. for m + n

n0,

In

particular,

^for

large n,

^we

have dn ⁽⁼ dn,o)

&#x3E; q ⁼ dim

Pic°X,

^the

number

given by Corollary (2.2).

^We claim that there exist elements of

~n(Sn(U)) ~ AW0(X) = A0(V)

^{which are not of the form}

(- 1)1-ICI(y)d

for any E9OE Pic X.

Indeed,

^{we have}

only

^to

verify

^that

U h - fi(Wi)

does not contain

~n(Sn(U)),

^{where h}

= A0(X) ~ AW0(X) is the

^natural

map. The ^set

is a

c-locally

^{closed set. We claim that}

which will _prove our earlier claim. Assume instead that

Up2(0393i,n) =

Sn(U).

^{Then some} irreducible

component W

of the c-closure of some

rl, n

^dominates

^Sn(U)

^{under the}

projection 0393i,n

^z

^{Sn(U). By}

definition of

d n ,

there exists ^a c-closed subset

F c ^Sn(U)

^{such that}

if x ~ Sn(U) - F,

then every irreducible

component

^{of the} c-closure of

~-1n(~n(x))

^has

dimension

nd - d n .

^{Thus if}

y ~ W

^{such that}

image ( y ) E Sn(U) - F,

then the fiber of

W ~ Wi through y

^has

dimension

nd -

d n .

^Hence

dim Wi

^{dim W -}

(nd - dn)

^dim

Sn(U) - nd + dn =d n

&#x3E; q ^{= dim}

Wi,

a contradiction.

Thus,

the theorem will be

proved

^{once we} show that there exists ^a

projective

A-module of rank d = dim A with ^a

prescribed top

^Chern class in

A0(V), V = Spec A,

^and

vanishing

lower Chern

classes,

^with

respect

^{to a suitable}

theory

of the Chow

ring

and Chern classes. For

non-singular ^V,

^{there is a standard}

theory (see [F]).

^We

briefly

^{review the}

properties

of Levine’s

theory [L1]

^{which are needed.}

First, CH1(V)

⁼

Pic V and

CHd(V) = A0(V) ^(the

^second

^equality

^{uses the}

^normality

^of

V).

^{The first} Chern class map

Ci

^satisfies

C1(Ox(D))

⁼

^{[D] ~} CH1(X)

as usual. If

P, Q

^are

projective ^modules, C(P), C(Q) ~ CH’(V)

^the

total Chern

classes,

^then

C( P

^ei

Q )

⁼

C(P), C(Q).

^The ’Riemann-Roch theorem without denominators’ is valid in the

following

^{sense: there is a}

filtration FiK0(V)

^and

maps 03A8i : CHi(F) ~ FiK0(V)/Fi+1K0(V)

^such

that if

Cl : FiK0(V)/Fi+1K0(V) ~ ^CHi(F)

is induced

by

the i th Chern class

Ci, then ’¥i 0 Cl

^and

Ci ° ’¥i

^{are both}

multiplication by (-1)i-1(i -

1) !.

^{For a}

subvariety

Z ~ V with Zn

Ving

⁼

^0,

^{codim Z =}

^i,

^{there is a}

class

[Z] ~ CHi(V)

^{such that}

Ci(OZ) = (-1)i-1(i-1)![Z]

^and

Cj(OZ)

= 0

for j i. Lastly,

^if

D1,...,Dd ~ V

^are effective Cartier divisors

meeting properly

^with

(~Di)

ⁿ

V5;ing ^{= ff,}

^then

[D1] · [D2]...[Dd] = [~Di ]

E

CHd(V)

⁼

A0(V).

^These

properties

suffice for our purpose.

We now show that

given

^any

03B4 ~ A0(V),

there exists ^a

projective

module P with rank P ⁼

d, C1(P) = ··· = Cd-1(P)

^{= 0 and}

Cd(P)

^{= 8.}

If I c A is the ideal of an effective

(reduced) 0-cycle supported

^on

V - Vsing,

then there is ^a resolution

where

F

^{are free and P’ is}

projective.

^Then

Ci(P’)

^{= 0 for i}

d,

^and

Cd(P’) = (-1)d-1Cd(A/I) = (d - 1)![A/I].

^{But any}

0-cycle

is ration-

ally equivalent

^{to a}

cycle

^{of the form}

(d - 1)![A/I ]

^{for an effective}

reduced

cycle [A/I],

since any ^zero

cycle

^{on V is}

supported

^{on a smooth}

affine curve C with C n

vsing

⁼

^{Ø, and}

^a similar claim is true for Pic C.

Thus we can find a

projective

^{module P with}

C;(P’) = 0

^for

i d,

Cd ( P’ )

= 03B4. Now if rank P’ = n

d, by

^the

stability

^{theorem of Bass}

[Ba]

^{we can write}

where P has rank d. Then

q(P)

^{= 0 for i}

^d, Cd ( P )

^{= 8.}

Applying

^this

construction to a

class 03B4 ~ ~n(Sn(U)) - ~ h ° fi(Wi)

^as

^above,

^{we see}

that P is

indecomposable.

This proves ^the

theorem.

§4. Examples

in dimension 2

The

examples

of affine varieties of

dimension

2 with nonzero Chow group of

0-cycles

^are

given by

^infinite

dimensionality

theorems for Chow groups of

projective

varieties. The best result is in dimension 2. Let X be

a normal

projective

^{surface over a universal} domain k. We say that

Ao(X)

^is

infinite

dimensional if none of the _maps

’ym,n

^of

§1

^are

surjective. Equivalently,

the number

d(X)

⁼

dR(X) given by Prop. (1.2)

^is

positive.

Then in fact

d(X) = 2, by

^Lemma

(1.3).

^{One can}

improve

lemma

(1.4)

^{as follows:}

d(X) = 0 ~ A0(X)’ ~

^{Alb X}

^(where A0(X)’

^~

AO(X)

^{is the}

subgroup generated by cycles

^of

degree 0).

^{Here Alb X is}

universal for rational maps from X ^to abelian

varieties;

ⁱⁿ

particular

^{it is}

a birational invariant. Thus Alb X = Alb Y for any resolution of

singu-

larities Y ~ X. In

fact,

^{one can show that}

d(X) = 0 ~ the composite A0(X) ~ A0(Y)’ ~ Alb

Y = Alb X is ^an

isomorphism (see [Sl],

^Theo-

rem

1).

^{But one}

easily

^sees that if U c X is an affine open ^set

containing

Xsing, ^{then A0(X)’ ~}

^Alb

^{Y ~ A0(U)}

^{= 0}

^(see

loc. cit.: the

point

^{is that}

X - U

supports

^an

ample divisor).

^In

particular, A0(U) ~ 0

^{if either}

A0(Y)’ ~

^{Alb Y}

(i.e. A0(Y)

is infinite

dimensional)

^or

A0(X) ~ A0(Y)

is not

injective.

For smooth surfaces in

arbitrary

characteristic

(generalising

^Mumford

[M2]

^{for k}

= C)

^Bloch

[Bl]

has shown that

A0(F)

is infinite dimensional if

and he also

conjectures

^{the converse. If k =}

C,

^{this is}

equivalent

^{to the}

condition

r( Y, 03A92Y) ~

^{0. If X is a normal}

projective surface,

^{it is shown}

in

[Sl] that A0(X) ~ A0(Y) is

^not

injective

^{in the}

following

^{cases. Let}

E c Y be the reduced

exceptional ^divisor,

^{which we may assume}

(blow

up ^Y

further)

has smooth

components

and normal

crossings.

^Then

(i)

^{If k}

= C,

^and

^{Pic0Y ~} Pic0(nE)

^{is nor}

surjective

^for some n &#x3E;

0,

then

A0(X) ~ A0(Y) is

^not

injective.

(ii)

^{if k is a} universal domain of

arbitrary characteristic,

^and

Pic0 Y ~ Pic0E

is not

surjective,

^then

A0(X) - A0(Y)

^{is not}

injective.

The condition

(i)

^is

equivalent

^{to the} condition that

H2(X, OX) ~ H2(Y, OY)

^{is not}

injective, by

^the

Leray spectral

sequence and the formal function theorem. Thus if k ⁼

C, A°(X)

^{is infinite} dimensional

provided H2(X, (9x) *

⁰

(we give

^another

proof

^{of this without K-the-}

ory in

[S2]).

^We

conjecture

^{that the converse}

^holds,

^{and that the converse}

to

(ii)

is also valid.

Assuming

^{the above}

conjectures

^{we have a}

fairly

^clear